Question

Differentiate the following functions w.r.t.x $fraction numerator cos left parenthesis log space straight x right parenthesis plus tan to the power of negative 1 end exponent straight x over denominator straight e to the power of straight x end fraction$

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Solution

Let space straight y equals fraction numerator cos left parenthesis log space straight x right parenthesis plus tan to the power of negative 1 end exponent straight x over denominator straight e to the power of straight x end fraction space space dy over dx equals fraction numerator straight e to the power of straight x. begin display style straight d over dx end style left curly bracket cos left parenthesis log space straight x right parenthesis plus tan to the power of negative 1 end exponent straight x right curly bracket minus left curly bracket cos left parenthesis log space straight x right parenthesis plus tan to the power of negative 1 end exponent straight x right curly bracket. begin display style straight d over dx end style left parenthesis straight e to the power of straight x right parenthesis over denominator left parenthesis straight e to the power of straight x right parenthesis squared end fraction space space space space space space space space space space equals fraction numerator straight e to the power of straight x open curly brackets begin display style fraction numerator negative sin left parenthesis log space straight x right parenthesis over denominator straight x end fraction end style plus begin display style fraction numerator 1 over denominator 1 plus straight x squared end fraction end style close curly brackets minus left curly bracket cos left parenthesis log space straight x right parenthesis plus tan to the power of negative 1 end exponent straight x right curly bracket. straight e to the power of straight x over denominator straight e to the power of 2 straight x end exponent end fraction space space space space space space space space space space equals fraction numerator begin display style fraction numerator negative left parenthesis 1 plus straight x squared right parenthesis sin left parenthesis log space straight x right parenthesis plus straight x minus straight x left parenthesis 1 plus straight x squared right parenthesis left curly bracket cos left parenthesis log space straight x right parenthesis plus tan to the power of negative 1 end exponent straight x right curly bracket over denominator straight x left parenthesis 1 plus straight x squared right parenthesis end fraction end style over denominator straight e to the power of straight x end fraction space space space space space space space space space space equals fraction numerator straight x minus left parenthesis 1 plus straight x squared right parenthesis sin left parenthesis log space straight x right parenthesis minus straight x left parenthesis 1 plus straight x squared right parenthesis space left curly bracket cos left parenthesis log space straight x right parenthesis plus tan to the power of negative 1 end exponent straight x right curly bracket over denominator straight x left parenthesis 1 plus straight x squared right parenthesis straight e to the power of straight x end fraction

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Continuity And Differentiability

Differentiate the following functions w.r.t.x $fraction numerator cos left parenthesis log space straight x right parenthesis plus tan to the power of negative 1 end exponent straight x over denominator straight e to the power of straight x end fraction$

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